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Mirrors > Home > MPE Home > Th. List > lmiclcl | Structured version Visualization version GIF version |
Description: Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
lmiclcl | ⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brlmic 20914 | . . 3 ⊢ (𝑅 ≃𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅) | |
2 | n0 4341 | . . 3 ⊢ ((𝑅 LMIso 𝑆) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) | |
3 | 1, 2 | bitri 275 | . 2 ⊢ (𝑅 ≃𝑚 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆)) |
4 | lmimlmhm 20910 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑓 ∈ (𝑅 LMHom 𝑆)) | |
5 | lmhmlmod1 20879 | . . . 4 ⊢ (𝑓 ∈ (𝑅 LMHom 𝑆) → 𝑅 ∈ LMod) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑅 ∈ LMod) |
7 | 6 | exlimiv 1925 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝑅 LMIso 𝑆) → 𝑅 ∈ LMod) |
8 | 3, 7 | sylbi 216 | 1 ⊢ (𝑅 ≃𝑚 𝑆 → 𝑅 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1773 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 class class class wbr 5141 (class class class)co 7404 LModclmod 20704 LMHom clmhm 20865 LMIso clmim 20866 ≃𝑚 clmic 20867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-1o 8464 df-lmhm 20868 df-lmim 20869 df-lmic 20870 |
This theorem is referenced by: lmisfree 21733 |
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